What is the acceleration of the particle the first time its velocity equals zero?
s(t)= -4sin(t) - (t/2) +10
What is the acceleration of the particle the first time its velocity equals zero?
−5.197
0.745
1.323
2.550
3.969
Could somebody write out the steps and explain this to me?
s(t)= -4sin(t) - (t/2) +10
v = ds/dt = -4cos t -1/2
a = d^2s/dt^2 = 4 sin t
what is t the first time v = 0?
4 cos t = -1/2
cos t = -1/8
t = 97.2 deg or 360 -97.2
it says the first time so
t = 97.2 deg * pi/180 = 1.7 radians
so what is a?
4 sin 97.2 = 3.97 looks like the last one
have to know how many time derivatives to take.
A particle moves along a line so that at time t, where 0 < t < π, its position is given by
Sorry! this was the first part
To find the acceleration of the particle when its velocity equals zero, we need to find the derivative of the given position function and then solve for the value of time (t) when the derivative equals zero.
Given position function: s(t) = -4sin(t) - (t/2) + 10
Step 1: Find the derivative of the position function
The first derivative of a function with respect to time (t) gives the velocity of the particle.
Derivative of s(t):
s'(t) = -4cos(t) - (1/2)
Step 2: Set the derivative equal to zero and solve for t
We need to find the value of t when the velocity is zero, so we set s'(t) = 0 and solve for t.
-4cos(t) - (1/2) = 0
Step 3: Solve for t
To isolate t, we can rearrange the equation:
-4cos(t) = (1/2)
cos(t) = -1/8
Now let's find the value of t between 0 and 2π at which cos(t) equals -1/8. To solve for t, we can use the inverse cosine function (arccos).
Step 4: Use inverse cosine to find t
t = arccos(-1/8) ≈ 1.907
So, the value of t at which the velocity equals zero is approximately 1.907.
Step 5: Find the acceleration
Now that we have the value of t, we can find the acceleration by taking the derivative of the velocity function:
a(t) = s''(t) = -4*(-sin(t)) = 4sin(t)
Substituting the value of t we found in Step 4:
a(1.907) = 4sin(1.907) ≈ 3.969
Therefore, the acceleration of the particle when its velocity first equals zero is approximately 3.969.